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Quantitative Aptitude

Progressions

Given a geome...

Question

Given a geometric progression
$a, a_{1}, a_{2}, a_{3}, . . . . . . . . .$
and an arithmetic progression
$b, b_{1}, b_{2}, b_{3}, . . . . . . . . .$
with positive terms. The common difference of A.P. and common ratio of G.P. are both positive. Show that there always exists a system of logarithms for which
$l o g a_{n} - b_{n} = l o g a - b$ (for any n)
Find base $β$ of the system

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Solution

Let r be the common ration of G.P. and d the common difference of A.P.
Then $a_{n} = a r^{n}$ ....(1)
and $b_{n} = b + n d$ .....(2)
Taking logarithms of both sides of (1) to base
$β (β \neq 1, β > 0)$ , we get
$l o g_{β} a_{n} = l o g_{β} a + n l o g_{β} r$
$l o g_{β} a_{n} - b_{n} = l o g_{b} a + n l o g_{β} r - b - n d .$ by (2).....(3)
Now in order that right hand side of (3) reduces to
$l o g_{β} a - b,$ we must have
$n l o g_{β} r - n d = 0$
or $l o g_{β} r = d o r r = β^{d}$ that is , $β = r^{1 / d}$
Hence there exists a system of logarithms to base $r^{1 / d}$
such that $l o g a_{n} - b_{n} = l o g_{n} a - b$

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